Optimal. Leaf size=31 \[ -\frac {a x}{b^2}+\frac {x^2}{2 b}+\frac {a^2 \log (a+b x)}{b^3} \]
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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {1598, 45}
\begin {gather*} \frac {a^2 \log (a+b x)}{b^3}-\frac {a x}{b^2}+\frac {x^2}{2 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 1598
Rubi steps
\begin {align*} \int \frac {x^4}{a x^2+b x^3} \, dx &=\int \frac {x^2}{a+b x} \, dx\\ &=\int \left (-\frac {a}{b^2}+\frac {x}{b}+\frac {a^2}{b^2 (a+b x)}\right ) \, dx\\ &=-\frac {a x}{b^2}+\frac {x^2}{2 b}+\frac {a^2 \log (a+b x)}{b^3}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 31, normalized size = 1.00 \begin {gather*} -\frac {a x}{b^2}+\frac {x^2}{2 b}+\frac {a^2 \log (a+b x)}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.35, size = 30, normalized size = 0.97
method | result | size |
default | \(-\frac {-\frac {1}{2} b \,x^{2}+a x}{b^{2}}+\frac {a^{2} \ln \left (b x +a \right )}{b^{3}}\) | \(30\) |
risch | \(-\frac {a x}{b^{2}}+\frac {x^{2}}{2 b}+\frac {a^{2} \ln \left (b x +a \right )}{b^{3}}\) | \(30\) |
norman | \(\frac {\frac {x^{3}}{2 b}-\frac {a \,x^{2}}{b^{2}}}{x}+\frac {a^{2} \ln \left (b x +a \right )}{b^{3}}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 29, normalized size = 0.94 \begin {gather*} \frac {a^{2} \log \left (b x + a\right )}{b^{3}} + \frac {b x^{2} - 2 \, a x}{2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.36, size = 29, normalized size = 0.94 \begin {gather*} \frac {b^{2} x^{2} - 2 \, a b x + 2 \, a^{2} \log \left (b x + a\right )}{2 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 26, normalized size = 0.84 \begin {gather*} \frac {a^{2} \log {\left (a + b x \right )}}{b^{3}} - \frac {a x}{b^{2}} + \frac {x^{2}}{2 b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.37, size = 30, normalized size = 0.97 \begin {gather*} \frac {a^{2} \log \left ({\left | b x + a \right |}\right )}{b^{3}} + \frac {b x^{2} - 2 \, a x}{2 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 29, normalized size = 0.94 \begin {gather*} \frac {2\,a^2\,\ln \left (a+b\,x\right )+b^2\,x^2-2\,a\,b\,x}{2\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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